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The Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality in the vacuum state of a relativistic scalar quantum field is revisited by making use of the Hilbert space HH₀₁, where H and H₀₁ stand, respectively, for the Hilbert space of the scalar field and of a generic bipartite quantum mechanical system. The construction of Hermitian, field-dependent, dichotomic operators is devised as well as the Bell-CHSH inequality. Working out the AB part of the inequality, the resulting Bell-CHSH correlation function for the quantum field naturally emerges from unitary Weyl operators. Furthermore, introducing a Jaynes-Cummings type Hamiltonian accounting for the interaction between the scalar field and a pair of qubits, the quantum corrections to the Bell-CHSH inequality in the vacuum state of the scalar field are evaluated till the second order in perturbation theory.
Guimarães et al. (Mon,) studied this question.